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Properties

Label 1-6145-6145.112-r1-0-0
Degree $1$
Conductor $6145$
Sign $-0.139 + 0.990i$
Analytic cond. $660.371$
Root an. cond. $660.371$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.617 + 0.786i)2-s + (−0.859 − 0.511i)3-s + (−0.238 − 0.971i)4-s + (0.932 − 0.360i)6-s + (−0.848 − 0.529i)7-s + (0.911 + 0.412i)8-s + (0.476 + 0.879i)9-s + (0.895 + 0.444i)11-s + (−0.292 + 0.956i)12-s + (0.826 − 0.563i)13-s + (0.940 − 0.340i)14-s + (−0.886 + 0.462i)16-s + (−0.984 − 0.178i)17-s + (−0.985 − 0.168i)18-s + (0.992 + 0.122i)19-s + ⋯
L(s)  = 1  + (−0.617 + 0.786i)2-s + (−0.859 − 0.511i)3-s + (−0.238 − 0.971i)4-s + (0.932 − 0.360i)6-s + (−0.848 − 0.529i)7-s + (0.911 + 0.412i)8-s + (0.476 + 0.879i)9-s + (0.895 + 0.444i)11-s + (−0.292 + 0.956i)12-s + (0.826 − 0.563i)13-s + (0.940 − 0.340i)14-s + (−0.886 + 0.462i)16-s + (−0.984 − 0.178i)17-s + (−0.985 − 0.168i)18-s + (0.992 + 0.122i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(660.371\)
Root analytic conductor: \(660.371\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (1:\ ),\ -0.139 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3188412047 + 0.3667441261i\)
\(L(\frac12)\) \(\approx\) \(0.3188412047 + 0.3667441261i\)
\(L(1)\) \(\approx\) \(0.5316084831 + 0.06003288065i\)
\(L(1)\) \(\approx\) \(0.5316084831 + 0.06003288065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (-0.617 + 0.786i)T \)
3 \( 1 + (-0.859 - 0.511i)T \)
7 \( 1 + (-0.848 - 0.529i)T \)
11 \( 1 + (0.895 + 0.444i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (-0.984 - 0.178i)T \)
19 \( 1 + (0.992 + 0.122i)T \)
23 \( 1 + (-0.262 + 0.964i)T \)
29 \( 1 + (-0.267 + 0.963i)T \)
31 \( 1 + (-0.997 + 0.0715i)T \)
37 \( 1 + (-0.542 - 0.840i)T \)
41 \( 1 + (0.257 - 0.966i)T \)
43 \( 1 + (-0.102 - 0.994i)T \)
47 \( 1 + (-0.471 + 0.881i)T \)
53 \( 1 + (-0.874 - 0.485i)T \)
59 \( 1 + (-0.316 - 0.948i)T \)
61 \( 1 + (0.355 + 0.934i)T \)
67 \( 1 + (0.995 + 0.0970i)T \)
71 \( 1 + (0.811 - 0.584i)T \)
73 \( 1 + (-0.993 - 0.117i)T \)
79 \( 1 + (-0.412 - 0.911i)T \)
83 \( 1 + (0.902 + 0.430i)T \)
89 \( 1 + (-0.502 + 0.864i)T \)
97 \( 1 + (-0.708 - 0.705i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.28960404945270653494656184204, −16.7432328261993722558879486101, −16.149075501203762003627431177402, −15.783164444862716838209786518561, −14.852781268536110971843169516131, −13.844916720169518154213061945149, −13.17152455026543043647120709739, −12.53340290858164192301667512242, −11.819035145908835340169454094573, −11.33148846443576596578171476127, −10.92109875061256624496643620613, −9.87289826414094611350708807046, −9.57764019030100845315928475854, −8.87728836202056384493894480102, −8.3471727030478056249234367807, −7.115431033564388231073634824470, −6.45885195091855685891647098748, −6.03156799903094476733160584901, −4.928296047526315639605139509906, −4.14048505912964493531973578515, −3.60759388039864053170759170835, −2.87828828158501801664914880916, −1.82571352057156616739922343920, −1.02356240101523553528259066847, −0.17292227733055699202221612669, 0.55118987845516665875860962346, 1.350104307308432741280074583303, 1.970070981788753322495566837004, 3.43137829047396332527436890038, 4.11449132750444796119879136132, 5.13882415536515094913888696924, 5.68080332935908459292041221218, 6.367628427480199399700891873360, 7.05153621460056972161937124060, 7.295913197994477803106705690216, 8.19712820420642361280045909153, 9.21102022893066886528764237766, 9.505790926391438988503809184782, 10.514960463867944534659297262962, 10.89492215610008961896958294954, 11.63471087428712945524509205467, 12.545755987483435851061802823076, 13.15263919940054327389035459419, 13.80515232582336969800844408197, 14.34671960495216765280887850065, 15.456808118348257969057433355670, 15.95422660356030296409239090124, 16.33502108248296329685666899359, 17.14336372532997795327607754175, 17.69980564620499015775483743123

Graph of the $Z$-function along the critical line